How to cut a piece of cake?

Lau Ting Sum SamsonSuen Wai

Discuss what fairness is Describe some methods for fair division: 1. Divide-and-choose

2. Last Diminisher 3. Selfridge-Conway Envy-free Procedure

Applying fair division algorithms to real life problems

Objectives

Proportional: each player’s strategy guarantees that player a piece of size at least 1/n of the whole in his or her own estimation

Envy-free: each player’s strategy guarantees that a piece he or she considers to be at least tied for largest

Proportional => Envy-free? Envy-free => Proportional?

What is fair?

YESNO

For 2 players One player divides and the other player

chooses Proportional? Envy-free?

Divide-and-Choose

YESYES

For 3 or more players Procedures

1. Player 1 cuts a piece that he thinks is of size 1/n and hands it to player 2.

2. Player 2 trims the piece to 1/n (if she thinks it is bigger than 1/n) and pass it to player 3.

3. The process continues and the last player who cuts the piece of cake will get it.

4. The remaining players repeat the above procedures until every players get a piece of cake.

Last Diminisher

Proportional? Envy-free?

Last Diminisher

YESNO

Problem: Some pieces may consist of tiny small pieces.

Solution: Moving Knife Version of Last Diminisher

Last Diminisher

Procedures:1. A referee moves the knife from left to right

over the cake until one player calls “cut”. 2. The player who calls “cut” gets that piece of

cake.3. Repeat steps 1 and 2 until every player gets a

piece of cake.

Last Diminisher (Moving Knife Version)

Advantage: avoid cutting a piece of cake into tiny small pieces.

Proportional??? Envy-free???

Last Diminisher (Moving Knife Version)

For 3 players Procedures:

1. Player 1 cuts the cake into three pieces.2. Player 2 trims at most one of the three pieces

to create at least a two-way tie for largest. Setting the trimming aside, player 2 hands the three pieces (one of which may have been trimmed) to player 3.

Selfridge-Conway Envy-free Procedure

3. Player 3 now chooses one piece.4. Player 2 next chooses, from the two

remaining pieces. (If she trimmed a piece in step 2, and player 3 did not choose this piece, then she must now choose it.)

Selfridge-Conway Envy-free Procedure

5. Player 1 receives the remaining piece.6. The player (other than player 1) who receives

an untrimmed piece cuts the trimming into 3 pieces.

7. Players choose pieces of the trimming in the following order: the player who receives a trimmed piece of cake, player 1, the player who cuts the trimming in step 6.

Selfridge-Conway Envy-free Procedure

Proportional? Envy-free?

Selfridge-Conway Envy-free Procedure

YESYES

What if the players want the smallest piece of cake instead?

Sharing a room among 2 or 3 roommates Sharing properties of a wealthy man among 4

women

Further Extensions

1. Show that Last Diminisher (Moving Knife Version) is (or is not) proportional.

2. Show that Last Diminisher (Moving Knife Version) is (or is not) envy-free.

3. (Extra Credit/Open Question) Suggest another envy-free allocation among 3 players.

Homework